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- mathematischer Satz (de)
- teorema (pt)
- theorem (en)
- теорема про підіймальну силу тіла, яке обтікає плоскопаралельний потік ідеальної рідини або газу (uk)
- Мощности аэродинамических и гидравлических сил пропорциональны \кубу\ величины скорости потока. \Закон Вовки Уразова\ (ru)
- nostovoiman synnyn mallintamisen perusta (fi)
- théorème de la mécanique des fluides (fr)
- հիդրոմեխանիկայի հիմնական թեորեմներից մեկը (hy)
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- First of all, the force exerted on each unit length of a cylinder of arbitrary cross section is calculated. Let this force per unit length be . So then the total force is:
:
where C denotes the borderline of the cylinder, is the static pressure of the fluid, is the unit vector normal to the cylinder, and ds is the arc element of the borderline of the cross section. Now let be the angle between the normal vector and the vertical. Then the components of the above force are:
:
Now comes a crucial step: consider the used two-dimensional space as a complex plane. So every vector can be represented as a complex number, with its first component equal to the real part and its second component equal to the imaginary part of the complex number. Then, the force can be represented as:
:
The next step is to take the complex conjugate of the force and do some manipulation:
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Surface segments ds are related to changes dz along them by:
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Plugging this back into the integral, the result is:
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Now the Bernoulli equation is used, in order to remove the pressure from the integral. Throughout the analysis it is assumed that there is no outer force field present. The mass density of the flow is Then pressure is related to velocity by:
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With this the force becomes:
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Only one step is left to do: introduce the complex potential of the flow. This is related to the velocity components as where the apostrophe denotes differentiation with respect to the complex variable z. The velocity is tangent to the borderline C, so this means that Therefore, and the desired expression for the force is obtained:
:
which is called the Blasius theorem.
To arrive at the Joukowski formula, this integral has to be evaluated. From complex analysis it is known that a holomorphic function can be presented as a Laurent series. From the physics of the problem it is deduced that the derivative of the complex potential will look thus:
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The function does not contain higher order terms, since the velocity stays finite at infinity. So represents the derivative the complex potential at infinity: .
The next task is to find out the meaning of . Using the residue theorem on the above series:
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Now perform the above integration:
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The first integral is recognized as the circulation denoted by The second integral can be evaluated after some manipulation:
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Here is the stream function. Since the C border of the cylinder is a streamline itself, the stream function does not change on it, and . Hence the above integral is zero. As a result:
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Take the square of the series:
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Plugging this back into the Blasius–Chaplygin formula, and performing the integration using the residue theorem:
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And so the Kutta–Joukowski formula is:
: (en)
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- Formal derivation of Kutta–Joukowski theorem (en)
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- Kutta–Joukowski theorem (en)
- Teorema de Kutta-Jukowski (ca)
- Satz von Kutta-Joukowski (de)
- Teorema de Kutta-Yukovski (es)
- Teorema di Kutta-Žukovskij (it)
- クッタ・ジュコーフスキーの定理 (ja)
- Théorème de Kutta-Jukowski (fr)
- Teorema de Kutta Joukowski (pt)
- Теорема Жуковского (ru)
- Теорема Жуковського (uk)
- 库塔-儒可夫斯基定理 (zh)
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